Mathematical Olympiad Treasures

Titu Andreescu Bogdan Enescu

Mathematical Olympiad Treasures
Second Edition

Titu Andreescu School of Natural Sciences and Mathematics University of Texas at Dallas Richardson, TX 75080 USA titu.andreescu@utdallas.edu

Bogdan Enescu Department of Mathematics “BP Hasdeu” National College Buzau 120218 Romania bogdanenescu@buzau.ro

ISBN 978-0-8176-8252-1 e-ISBN 978-0-8176-8253-8 DOI10.1007/978-0-8176-8253-8 Springer New York Dordrecht Heidelberg London
Library of Congress Control Number: 2011938426 Mathematics Subject Classification (2010): 00A05, 00A07, 05-XX, 11-XX, 51-XX, 97U40 © Springer Science+Business Media, LLC 2004, 2011 All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (SpringerScience+Business Media, LLC, 233 Spring Street, New York, NY 10013, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use in this publication of trade names, trademarks,service marks, and similar terms, even if they are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights. Printed on acid-free paper Springer is part of Springer Science+Business Media (www.birkhauser-science.com)

Preface

Mathematical Olympiads have a tradition longer than one hundred years. The first mathematicalcompetitions were organized in Eastern Europe (Hungary and Romania) by the end of the 19th century. In 1959 the first International Mathematical Olympiad was held in Romania. Seven countries, with a total of 52 students, attended that contest. In 2010, the IMO was held in Kazakhstan. The number of participating countries was 97, and the number of students 517. Obviously, the number of young studentsinterested in mathematics and mathematical competitions is nowadays greater than ever. It is sufficient to visit some mathematical forums on the net to see that there are tens of thousands registered users and millions of posts. When we were thinking about writing this book, we asked ourselves to whom it will be addressed. Should it be the beginner student, who is making the first steps in discoveringthe beauty of mathematical problems, or, maybe, the more advanced reader, already trained in competitions. Or, why not, the teacher who wants to use a good set of problems in helping his/her students prepare for mathematical contests. We have decided to take the hard way and have in mind all these potential readers. Thus, we have selected Olympiad problems of various levels of difficulty. Some arerather easy, but definitely not exercises; some are quite difficult, being a challenge even for Olympiad experts. Most of the problems come from various mathematical competitions (the International Mathematical Olympiad, The Tournament of the Towns, national Olympiads, regional Olympiads). Some problems were created by the authors and some are folklore. The problems are grouped in three chapters:Algebra, Geometry and Trigonometry, and Number Theory and Combinatorics. This is the way problems are classified at the International Mathematical Olympiad. In each chapter, the problems are clustered by topic into self-contained sections. Each section begins with elementary facts, followed by a number of carefully selected problems and an extensive discussion of their solutions. At the end of eachsection the reader will find a number of proposed problems, whose complete solutions are presented in the second part of the book.
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Preface

We encourage the beginning reader to carefully examine the problems solved at the beginning of each section and try to solve the proposed problems before examining the solutions provided at the end of the book. As for the advanced reader, our...